For shipments $x_{ij}\ge 0$ with factory row sums at most capacities $s_i$ and shop column sums exactly demands $d_j$, the displayed dual potentials give $\sum c_{ij}{x}_{ij}\ge {\sum }_i{u}_i{s}_i+{\sum }_j{v}_j{d}_j$. The sign $u_i\le 0$ ensures $u_i{\sum }_j{x}_{ij}\ge u_i{s}_i$. Equality proves optimality. It occurs when every used cell is tight and every row with spare capacity has $u_i=0$, the relevant complementary slackness conditions. A dummy demand column with zero costs converts unused capacity into a balanced transportation problem.

A finite word over an alphabet is locally admissible for a zero-one symbolic transition matrix if every consecutive pair is permitted. Its length counts symbols, so length $n+1$ corresponds to $n$ transitions. The number with endpoints $i,j$ is $(A^n{)}_{ij}$ by matrix multiplication. Local admissibility does not always imply occurrence in a two-sided subshift of finite type: the matrix $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ permits the word $(1,2)$ but admits no two-sided sequence. If every row and column has a nonzero entry, every locally admissible word extends in both directions.

A complex vector bundle has structure group $G$ when it admits fiber-linear local trivializations whose transition matrices take values in $G$. Replacing frames by $h_i$ changes transitions to $h_i^{-1}{g}_{ij}{h}_j$ in the frame convention. Reducing the structure group means choosing new trivializations with values in a specified subgroup. The same transitions glue $U_i\times G$ by left multiplication to a right principal bundle.

Prediction optimism is the difference between expected squared prediction error on new independent responses at the training covariates and expected training error. Reusing responses to fit and assess a model tends to make training error too favorable. The covariance formula for prediction optimism measures this reuse without assuming a linear fitting rule.

// Target: probability-and-statistics.bigb

For a trace-class operator $K$ on a Hilbert space, the Fredholm determinant is the convergent product over its eigenvalues, counted with algebraic multiplicity. For self-adjoint $K$, spectral theorem for compact self-adjoint operators reduces the product to a diagonal calculation. For $\Vert K\Vert <1$, $\log \det (I+K)={\sum }_{n\ge 1}(-1{)}^{n+1}\mathrm{tr}(K^n)/n$. This identity exposes the successive cumulants in a Gaussian quadratic exponential moment.

The dual of the trace-class operators, and also of the normed space of finite-rank operators with the trace norm, is isometrically the space of bounded operators. The pairing is $F_B(T)=\mathrm{Tr}(BT)$. For $R_{x,y}z=⟨z,y⟩x$, one has $\Vert R_{x,y}{\Vert }_1=\Vert x\Vert \Vert y\Vert$ and $F_B(R_{x,y})=⟨Bx,y⟩$, using an inner product linear in its first argument. A bounded functional on these rank-one operators gives a bounded sesquilinear form and hence a unique $B$ by the Riesz representation theorem. The singular value decomposition proves $\Vert F_B\Vert =\Vert B\Vert$ and the triangle inequality for the trace norm.

If $L/K$ is an unramified extension and $M/K$ is a totally ramified extension of non-Archimedean local fields inside a fixed algebraic closure, then $LM/L$ is totally ramified of degree $[M:K]$. A uniformizer of $M$ has an Eisenstein polynomial over $K$. Since a uniformizer of $K$ remains a uniformizer in $L$, that polynomial remains Eisenstein over $L$ and generates $LM$.

A quadratic form of rank $r$ on ${\mathbb{C}}^n$ is congruent to $x_1^2+\cdots +x_r^2$. Pair its nonzero coordinates into vectors $e_{2j-1}+i{e}_{2j}$, and include the radical of a bilinear form. Their span is a totally isotropic subspace and has dimension $\lfloor r/2\rfloor +n-r=n-\lceil r/2\rceil \ge \lfloor n/2\rfloor$. The form vanishes on every vector of this subspace, not merely on its basis vectors.

For a nondegenerate real quadratic form of inertia $(p,q)$, projection of a totally isotropic subspace onto either its positive or negative coordinate subspace is injective: a vector with one projection zero would have strictly one-signed quadratic value unless it vanished. Thus its dimension is at most both $p$ and $q$. This bound is attained by pairing one positive and one negative coordinate for each of $\min (p,q)$ independent null vectors.

For a Pareto distribution of shape three and minimum $d>0$ with count rate $\lambda >0$, the sum of insurer and reinsurer aggregate variances is $V(M)=\lambda (3d^2-3dM+2M^2)$ for $0\le M\le d$, and $\lambda (3d^2-d^3/M)$ for $M\ge d$. The first branch has its unique minimum at $3d/4$; the second is increasing. The branches agree in value and slope at $d$, giving the displayed global minimum. This criterion minimizes the sum of party variances; the total portfolio variance includes their covariance and remains $3\lambda d^2$.

Two disjoint synthemes form a six-cycle, whose complement is a triangular prism. The prism has a unique partition into three perfect matchings, so the pair extends uniquely to a total of synthemes. Eight synthemes are disjoint from a fixed syntheme; grouping them four per total yields two totals through it, and double counting gives six totals in all.

// Target: combinatorics.bigb

A path along a circle can have variable areal velocity. As an illustrative controlled trajectory, keep a point mass on a circle of radius $R$ with angleIts acceleration in polar coordinates and the required net force arewhere ${\mathbf{e}}_r$ and ${\mathbf{e}}_{\theta }$ are the radial and transverse unit vectors. A circular guide can supply the radial force, while a tangential drive supplies $mR\alpha$. HenceFor $R=m={\omega }_0=\alpha =1$ in illustrative units, the three equal time intervals $[0,0.5]$, $[0.5,1]$ and $[1,1.5]$ sweep areas $0.3125$, $0.4375$ and $0.5625$. Their difference comes from the transverse force's positive torque, even though the radius and path shape stay fixed. The area under each areal velocity graph segment equals the corresponding swept area.

An axial uniform magnetic field and localized radial differential rotation generate a steady toroidal magnetic field. The angular factor $\sin \theta \cos \theta$ is an eigenfunction of the toroidal angular diffusion operator with eigenvalue $-6$. Its radial amplitude satisfies $\eta (f^{\prime \prime }+2f^{\prime }/r-6f/r^2)=-B_{0}r{\Omega }^{\prime }$. Differentiating the integral in the title proves the formula. It is regular at the origin and decays at infinity when $\Omega$ is finite at zero and decays sufficiently rapidly.

A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by $[X,Y{]}_{\ast }$ for a CW complex $Y$. Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.

A single topological generator of a topological group $G$ is an element $g$ whose cyclic subgroup is dense:More generally, a tuple $g_1,\dots ,g_d$ consists of topological generators when the subgroup they jointly generate is dense. Membership in such a tuple does not imply that one element alone generates densely.

In the additive group ${\mathbb{Z}}_p$ of p-adic integers, the element one is a topological generator: ordinary integers surject onto every finite quotient $\mathbb{Z}/p^n\mathbb{Z}$. The element $p$ generates only the proper closed subgroup $p{\mathbb{Z}}_p$. The profinite group ${\mathbb{Z}}_p^2$ has a two-element topological generating set but no single topological generator, since its quotient $(\mathbb{Z}/p\mathbb{Z}{)}^2$ is not cyclic. ^[ref]

An isotonic solution produces no sustained change in cellular volume under the stated comparison conditions. Water can still exchange in both directions.

A hypertonic solution favors initial net water loss under the reference conditions above. In a walled plant cell, enough loss can produce plasmolysis.

A hypotonic solution favors initial net water entry under the reference conditions above. A susceptible biological cell may undergo osmotic lysis; a sufficiently strong cell wall can instead support turgor pressure.

A permeating solute can cross the specified membrane appreciably during the observation interval. This is a property of a solute-membrane-time comparison, not an immutable label attached to a molecule. Passage can reduce a concentration difference through passive transport while water is also moving; it does not require water to wait.

For an original dilute ideal example, stipulate a flexible model biological cell with $20\mathrm{mmol}{\mathrm{L}}^{-1}$ retained particles and an external reservoir with $10\mathrm{mmol}{\mathrm{L}}^{-1}$ retained particles plus $10\mathrm{mmol}{\mathrm{L}}^{-1}$ freely permeating neutral solute. Initially the total osmolarities are equal. Suppose the permeating solute eventually has equal internal and external concentrations, there is negligible pressure difference, no active transport, and the internal retained particle amount is conserved. The permeating contributions cancel from the final osmotic balance; the internal retained concentration must fall from $20$ to $10\mathrm{mmol}{\mathrm{L}}^{-1}$. The required volume is twice its initial value. This illustrates an initially equal-osmolarity but hypotonic solution; it assumes the boundary can survive that expansion and predicts no universal time course.

For $H(t)={\Omega }_1(t)A+{\Omega }_2(t)B$ with fixed noncommuting Hermitian operators, $[H(t),H(s)]=[{\Omega }_1(t){\Omega }_2(s)-{\Omega }_2(t){\Omega }_1(s)][A,B]$. A constant ratio of the envelopes is sufficient to remove time ordering. Independently changing that ratio generally produces commutator corrections in the Magnus expansion, and a product of successive pulses need not equal the exponential of their summed areas.